To establish the rate of convergence for Algorithm  in Theorem REF , we first present Lemma REF  and Lemma REF  regarding the basic recursion on the errors in estimating the inner function values and the order of \(\mathbb {E}[\Vert  u_{i}^{k+1} - u_{i}^{k}\Vert ^2|{F}_k]\) . The proofs follow [1]} with minor modifications. We present the complete proofs below for the reader's convenience.
